Introduction
From personal experience of learning mathematics throughout my academic life, I have found that mathematics is best taught if taught as a way to solve problems. Based on my experiences in studying and teaching mathematics, I have come to subscribe to the fallibilist philosophy of mathematics which entails envisioning mathematics as an outcome of various social processes (White-Fredette, 2009). I understand mathematics to be fallible, and its proofs and concepts are always open to revision. This perception of mathematics lets me recognize the role that mathematics plays in human culture, especially in issues of value and, more importantly, education. Based on this, I believe that students should experience mathematics not only as a pure body of abstract knowledge but as human, personal, intuitive, and creative, making it enjoyable and joyful for the student. This allows students to understand the underlying concepts better. Zain, Rasidi & Abidin (2012) point out that the best way to make it so is to teach mathematics as a problem-solving skill that is applicable in the classroom and a student’s personal and social life. This essay recommends teaching mathematics as a problem-solving tool and justifies it by analyzing personal experiences through mathematics education ideas drawn from class readings.
Problem Solving in Mathematics
As a student, I usually found that the role of the teacher was not to instruct a student but to help them. Although this may take time, commitment, and practice for both the teacher and the student, the best way for the student to learn is to gain experience from their own autonomous work as much as possible. However, this requires a balance as if the student is left to fend for themselves in the learning process; they may not make substantial progress. On the other hand, if the teacher is overly involved in the students learning process, the student has no independent experience to gain. Therefore, the teacher should aim not to offer support too much or too little so that the student has a chance to learn by themselves as well as with the help of the teacher (Biggs & Tang, 2011). This principle applies when teaching mathematics as a way of problem-solving. In my understanding of the fallibilist philosophy of mathematics, a problem in mathematics is a task with which the student has no memorized methodology and no abstract idea of a correct solution (Brown & Coles, 2012). Based on this, problem-solving in mathematics becomes the process by which the student identifies the mathematical problem, tries to identify varying alternatives for a solution, and evaluates the conclusions they arrive at.
I have found that learning by problem-solving is the best way to teach mathematics. It allows the student to figure out ways to solve problems in their life. Yuanita, Zulnaidi & Zakaria (2018 ascertain that in the process of understanding problem-solving in a mathematical context, the student also gains a more profound and broader comprehension of mathematical concepts. This teaching method helps the student understand that learning and memorizing mathematical concepts is not enough; it is equally essential for the student to learn how to utilize these facts to enhance their thinking. Therefore, problem-solving entails mathematical activities that can intellectually challenge the student as it enhances the learner’s understanding of mathematical concepts (Simamora & Saragih, 2019). The teacher should teach mathematics as a variety of problems that enable students to gain contexts that will be crucial in their development in mathematics. This way, the student can learn to think instead of aimlessly cramming and memorizing formulas and concepts.
Problem Solving and Teaching
There are three main ways a teacher can educate a student to solve problems in mathematics. The first is to teach problem-solving. In this method, the student starts by learning a mathematical skill. An example is when the students are learning multiplication, e.g., multiplying a three-digit number with a two-digit number, the teacher needs to use story problems that are multiplication problems that they are planning to teach. This way, the student can learn the mathematical concepts of multiplication through problem-solving as the teacher selects questions that also promote the students’ understanding of mathematics. The second method is to teach the student in detail about problem-solving. This would involve teaching the student a problem-solving methodology where the student starts by reading the problem, then devises a plan on how to solve the problem, then solve the problem and finally check and verify the accuracy of their work. This way, when the student encounters a mathematical problem such as a word or story problem, they focus on utilizing the steps in the problem-solving methodology rather than the mathematical concept itself. This way, the student can focus on making sense of the problem rather than use trial and error to solve it. Finally, the teacher needs to teach the student through problem-solving. Through this method, the student pays attention to the underlying concepts and makes sense of them, thereby improving their mathematical practices. Teaching through problem-solving enables the student to gain confidence in their understanding of mathematical concepts and enhance their strengths while at the same time recognizing what they do not comprehend, allowing them to make an effort to improve. Through these methods of teaching problem solving, the students can easily collaborate and make them more engaging in their learning of mathematical concepts.
Criteria for Worthwhile Problems
For a problem to enhance the student’s problem-solving skills, it must abide by certain criteria. The problem needs to be vital and contain useful mathematical concepts within it. It must also require critical thinking and problem solving while at the same time allowing the students to develop their mathematical concepts. The problem must also encourage the student to be more engaging and participate in a personal and group discussion on the concepts within. It should encourage the student to use mathematical ideas skillfully and enable the student to practice these skills. A problem that meets these criteria allows the teacher to assess the student’s learning to identify their strengths and recognize where they do not comprehend so that they can offer assistance. Rationally speaking, not every problem will meet all of the above criteria. In some instances, the teacher might offer the student a problem that does not abide by these criteria so that they can have the opportunity to practice their skills. Bottomline, the problem issued to the student to solve must be based on the concepts that the students need to learn or those that they already understand. Whimbey, Lockhead & Narode (2013) affirm that to foster problem-solving and critical thinking; the teacher must insist that the student offer an explanation of the answers they come up with and the method they used to solve them.
Mathematical Activities that Enable Teaching through Problem Solving
There are several activities that a teacher can utilize when using problem-solving to teach mathematics. The first is by using the low floor, high ceiling tasks. These are tasks where every student in the class begins solving the task, and they then move on to work on the task based on their level of engagement (Hughes, Gadanidis & Yiu, 2017). In these types of tasks, the path to finding a solution is more important than the solution itself, prompting the students to engage in meaningful mathematical discussion. A teacher can also use the Maths in 3-Acts task, which invokes interest and enhance the student’s engagement in a thought-provoking process of mathematical inquiry (Redmond-Sanogo et al., 2018). This approach consists of 3 acts. The first act involves noticing and sparking hunger. The teacher introduces an engaging situation, and the students come up with questions regarding the situation. The second act involves the teacher providing the students with information which the students use to come up with solutions to the problem. The third act involves the students sharing their line of thought and providing the solutions they come up with. These approaches encourage students to engage in critical discussions and strengthen their confidence in their abilities.
Conclusion
A teacher’s method of teaching mathematics is primarily based on their perception of the subject. As a fallibilist, I find mathematics to have its place in human culture; therefore, its teaching must be based on more than pure abstract knowledge but also its application in a student’s life. Consequently, I recommend that mathematics be taught as a way of problem-solving. The teacher must dedicate themselves to helping the students learn rather than plainly instructing them. Teaching mathematics through problem-solving allows the student to understand that mathematics is more than just learning facts; it also involves learning how to use these facts to develop their thinking. The teacher can utilize three ways to teach students to solve problems in the context of mathematics, namely, teaching for problem-solving, teaching about problem-solving, and teaching through problem-solving. They can accomplish this through activities such as the low floor, high ceiling and Maths in 3-Acts. Through this method of education, students become more engaged in their learning.
Reference
Biggs, J., & Tang, C. (2011). Teaching for quality learning at university. McGraw-hill education (UK). https://books.google.com/books?hl=en&lr=&id=VC1FBgAAQBAJ&oi=fnd&pg=PP1&dq=,+the+teacher+should+aim+not+to+offer+support+too+much+nor+too+little+so+that+the+student+has+a+chance+to+learn+by+themselves+as+well+as+with+the+support+of+the+teacher&ots=E8BJsCeBNo&sig=j83K2WCgFvWK57PRj077nSkllAw
Brown, L., & Coles, A. (2012). Developing “deliberate analysis” for learning mathematics and for mathematics teacher education: how the enactive approach to cognition frames reflection. Educational Studies in Mathematics, 80(1), 217-231. https://link.springer.com/article/10.1007/s10649-012-9389-7
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Simamora, R. E., & Saragih, S. (2019). Improving Students’ Mathematical Problem Solving Ability and Self-Efficacy through Guided Discovery Learning in Local Culture Context. International Electronic Journal of Mathematics Education, 14(1), 61-72. https://eric.ed.gov/?id=EJ1227360
Whimbey, A., Lochhead, J., & Narode, R. (2013). Problem solving & comprehension. Routledge. https://www.taylorfrancis.com/books/mono/10.4324/9780203130810/problem-solving-comprehension-arthur-whimbey-jack-lochhead-ron-narode
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Yuanita, P., Zulnaidi, H., & Zakaria, E. (2018). The effectiveness of Realistic Mathematics Education approach: The role of mathematical representation as mediator between mathematical belief and problem solving. PloS one, 13(9), e0204847. https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0204847
Zain, S. F. H. S., Rasidi, F. E. M., & Abidin, I. I. Z. (2012). Student-centred learning in mathematics constructivism in the classroom. Journal of International Education Research (JIER), 8(4), 319-328. https://clutejournals.com/index.php/JIER/article/view/7277
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